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Alex J. Loosley,Xian M. OBrien,Jonathan S. Reichner,Jay X. Tang
Many cell types can bias their direction of locomotion by coupling to external cues. Characteristics such as how fast a cell migrates and the directedness of its migration path can be quantified to provide metrics that determine which biochemical and biomechanical factors affect directional cell migration, and by how much. To be useful, these metrics must be reproducible from one experimental setting to another. However, most are not reproducible because their numerical values depend on technical parameters like sampling interval and measurement error. To address the need for a reproducible metric, we analytically derive a metric called directionality time, the minimum observation time required to identify motion as directionally biased. We show that the corresponding fit function is applicable to a variety of ergodic, directionally biased motions. A motion is ergodic when the underlying dynamical properties such as speed or directional bias do not change over time. Measuring the directionality of nonergodic motion is less straightforward but we also show how this class of motion can be analyzed. Simulations are used to show the robustness of directionality time measurements and its decoupling from measurement errors. As a practical example, we demonstrate the measurement of directionality time, step-by-step, on noisy, nonergodic trajectories of chemotactic neutrophils. Because of its inherent generality, directionality time ought to be useful for characterizing a broad range of motions including intracellular transport, cell motility, and animal migration.
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Describing Directional Cell Migration with a Characteristic Directionality Time
May Describing Directional Cell Migration with a Characteristic Directionality Time Alex J. Loosley 0 1 2 Xian M. OBrien 0 1 2 Jonathan S. Reichner 0 1 2 Jay X. Tang 0 1 2 0 1 Department of Physics, Brown University , Providence, RI , USA , 2 Division of Surgical Research, Department of Surgery, Rhode Island Hospital and the Warren Alpert Medical School of Brown University , Providence, RI , USA 1 Funding: This work was partially supported by National Science Foundation grant PHYS-1058375 (to JXT), National Institutes of Health grants GM- 066194 and AI-079582 (to JSR), allocations to the Department of Surgery by Rhode Island Hospital, and by a Natural Sciences and Engineering Research Council of Canada Post Graduate Fellowship (to AJL). The funders had no role in study design , data 2 Academic Editor: Maddy Parsons, Kings College London, UNITED KINGDOM Many cell types can bias their direction of locomotion by coupling to external cues. Characteristics such as how fast a cell migrates and the directedness of its migration path can be quantified to provide metrics that determine which biochemical and biomechanical factors affect directional cell migration, and by how much. To be useful, these metrics must be reproducible from one experimental setting to another. However, most are not reproducible because their numerical values depend on technical parameters like sampling interval and measurement error. To address the need for a reproducible metric, we analytically derive a metric called directionality time, the minimum observation time required to identify motion as directionally biased. We show that the corresponding fit function is applicable to a variety of ergodic, directionally biased motions. A motion is ergodic when the underlying dynamical properties such as speed or directional bias do not change over time. Measuring the directionality of nonergodic motion is less straightforward but we also show how this class of motion can be analyzed. Simulations are used to show the robustness of directionality time measurements and its decoupling from measurement errors. As a practical example, we demonstrate the measurement of directionality time, step-by-step, on noisy, nonergodic trajectories of chemotactic neutrophils. Because of its inherent generality, directionality time ought to be useful for characterizing a broad range of motions including intracellular transport, cell motility, and animal migration. – Directional cell migration is the process in which a single cell or a group of cells bias their direction of locomotion by coupling to an external cue. External cues may be soluble in nature such as during chemotaxis , insoluble such as during haptotaxis , or mechanical such as during durotaxis  and gravitaxis . Processes involving directional cell migration are ubiquitous in nature and essential for many fundamental biological processes facilitating the innate and adaptive immune systems [5, 6], sexual reproduction , embryonic development , cancer metastasis [9, 10], and more. The efficacy to which cells are able to carry out these functions is often tied to the characteristics of their migration, including migration speed, persistence, and collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. tortuosity . These characteristics can be quantified to determine which biochemical and biomechanical factors affect cell migration, and by how much, but choosing the right metric is important. To introduce this work, we briefly review several commonly used metrics for characterizing cell migration to show that the current paradigm is good for characterizing persistent motion in the absence of an external cue, but does not reproducibly characterize directional motion. The main goal of this work is to derive an intuitive metric for reproducibly quantifying the directional bias of motion, independent of persistence. Commonly Used Analytical Tools For Characterizing Migration Mean squared displacement (MSD) is a common metric for measuring migration speed and distance traveled because it is easily interpretable and readily derived from mathematical models of motion. Numerous studies that characterize directional migration use MSD in conjunction with at least one other metric for quantifying path persistence or tortuosity . Three examples of such metrics used are: the distribution of turning angles between discrete measurements of centroid displacements (turning angle distribution, TAD); tortuosity (also known as straightness index [18, 19], chemotactic index [15, 20], or directionality ratio ) defined as the end-to-end distance traveled divided by the total migration path length; and tangent-tangent correlation, which describes the correlation in migration path orientation over a specified length or time interval. Sampling Interval Dependence In order to gain insight from quantitative characterizations of the migration path, the numerical values of the metrics applied must be reproducible from one set of experiments to another. Such values should also reflect the true kinematic properties of migration by decoupling from pseudo random kinematics induced by measurement error along the migration path. One common shortcoming of TAD, tortuosity, and tangent-tangent correlation is that they each implicitly depend on sampling interval, t, which is the time interval between position measurements. Sampling interval can be chosen arbitrarily, implying that TAD, tortuosity, and tangent-tangent correlation curves are not generally reproducible without using an equivalent sampling interval across all experiments. Even when sampling intervals are accounted for, a sampling interval dependent metric only characterizes migration at an arbitrarily chosen time scale at which the observable may or may not decouple from measurement error. To visualize sampling interval dependence, consider two experimental measurements of a migration path, one using a long sampling interval, t = t, and the other using a short sampling interval, t = t (Fig 1A, top and bottom, respectively). Circles are centroid positions and the corresponding perceived migration paths (red line segments connecting circles) are juxtaposed against the true migration path (thick grey curve). The deviation between centroid positions (rn = r(tn), where tn = nt, n = 1, 2, . . ., N, t = t or t) and the true migration path represents centroid measurement error, which depends on factors such as image resolution and cell boundary detection accuracy. Angles between successive red line segments, n, are turning angles ( ). Taking into account all turning angles, the normalized TAD, n(; t), is calculated for both sampling intervals (Fig 1B). As the sampling interval increases towards the total duration of the migration path, the TAD curve becomes sharply peaked at = 0. Conversely, as the sampling interval decreases towards zero, the effects of diffusive motion and centroid measurement error flatten the TAD curve. Hence, TAD depends notably on sampling interval. One measure of persistence is the so-called turning angle persistence (TAD persistence), the fraction of all turning angles between p2 (shaded area under TAD curves in Fig 1B). TAD persistence depends on the sampling interval just as TAD does. Fig 1. The effects of sampling interval on measurements and characterization of migration trajectories. (A) A migration path sampled with a long sampling interval, t = t (top, outlined yellow circles), and a short sampling interval, t = t (bottom, outlined white circles). Using the long sampling interval diagram, the observed migration path is formed by connecting measurements of centroid positions, r1, r2, . . ., with lines (in red). Unit tangent vectors are shown as v^1; v^2; ::: while turning angles, 1, 2, . . ., are defined as the angle between successive tangent vectors. Differences in centroid position from the true migration path (thick grey curve) represent measurement error. (B) Turning angle distributions (TAD), n(;t), based on both the long (top) and short (bottom) sampling intervals. A measure of migration persistence known as TAD persistence is the area under the TAD curve between p2 (shaded). TAD persistence depends notably on sampling interval. Similar diagrams can be used to show the sampling interval dependence of metrics such as tortuosity and tangent-tangent correlation. (C) A diagram visually defining terms necessary to analytically derive the directionality time model. A cell is depicted as a random walker with steps (R1, R2, . . ., Rn, . . .) located in a 2-D coordinate system defined by the unit vectors e^x and e^y. Capital letters correspond to random variables. For each step, there is a corresponding step length Ln and polar angle n. The latter is defined with respect to unit vector e^x and should not be confused with the corresponding turning angle, n = n n1. The dependence of tortuosity on sampling interval is apparent when considering the limit that the sampling interval approaches the total duration of the migration path. In this limit, total path length is the end-to-end length resulting in a tortuosity of 1. When the sampling interval decreases, the total path length increases due to centroid measurement error and the underlying fractal nature of the migration path itself [19, 22, 23]. Sampling Interval Independent Metrics Tangent-tangent correlation, v^t t v^t, is the time averaged cosine of the angle between tangent vectors v^t t and v^t that are separated by a time interval . The overline denotes an average over all time t. When applied to discrete experimental data captured at a specific sampling interval, v^t is replaced by v^n (Fig 1A). Tangent-tangent curves are sampling interval dependent because the difference between the true instantaneous tangent vector and the measured tangent vector generally increases as the sampling interval increases. However, there is a sampling interval independent measure called persistence time that derives from the tangent-tangent correlation curve. Persistence time, tp, is the time scale below which directional orientation of the migration path remains correlated. In general, persistence time is measured as a fit parameter in a model that fits the tangent-tangent correlation curve over all time scales. Therefore, measurements of persistence time are sampling interval independent so long as the sampling interval is small enough to provide enough data points for a good fit. Persistence time and migration speed together are sufficient to characterize non-directional motion but an additional metric is needed when motion is directional (biased by an external cue). This third metric should be sampling interval independent, like persistence time, and decouple from measurement error. Such a metric is derived in this article. Sampling interval independence implies an integration of data over all time intervals rather than choosing to make a measurement based on one specific sampling interval. One approach to achieving a sampling interval independent metric of directionality could entail fitting data such as TAD persistence or tortuosity to a model, thereby measuring a set of fit parameters. However, TAD persistence and tortuosity models are difficult to calculate and interpret. MSD models are easier to calculate and interpret and several models have already been analytically derived [19, 24, 25]. Nevertheless, this approach of fitting to a model only works if the underlying kinematics of the migration are understood a priori such that a model can be chosen. While one can attempt to fit more than one model to determine which fits best, changes to MSD from one model to the next can be small with respect to the error bars on an experimental MSD curve. Hence, there is the possibility of a causality loopone cannot accurately understand a set of migration paths without knowledge of the underlying process and corresponding random walk model, but one is unsure of the corresponding random walk model without understanding the migration paths. To circumvent this causality loop, we take a bottom up approach to derive a generalized sampling interval independent metric called directionality time. Whereas persistence time is the time scale below which the orientation of the migration path remains correlated, directionality time is defined as the time scale above which the orientation of the migration path becomes correlated due to an external directional bias. Directionality time supplements persistence time when characterizing directional random walk-like motion. We derive a directionality time fit model that is broadly applicable to many types of ergodic directional motion and decouples from measurement error. We also discuss how the directionality time fit model can be adapted to handle heterogeneous populations of random walkers and nonergodic motion. Finally, we demonstrate its implementation on data of chemotactic neutrophils that migrate directionally with non uniform speed. This section contains three subsections reviewing ensemble averaged squared displacement, mean squared displacement (MSD), and ergodicity, followed by two subsections with general methods information. The concepts reviewed here are necessary to conceptually derive directionality time. The models contained within this article are derived in full mathematical detail in the supporting information (Appendices A and B in S1 File). Mean Squared Displacement We begin with the assertion that ensemble averaged squared displacement follows a power law where i = 1, 2, 3, . . . is the index of one migration path in an ensemble, and the angle brackets hi denote the ensemble average over squared displacements measured at time t. To be precise about the type of averaging, we call this quantity the ensemble averaged squared displacement (EASD) instead of MSD. The exponent characterizes the motion. A constant value of = 1 indicates diffusive (random) motion whereas = 2 indicates ballistic (directional) motion. Other values represent subdiffusive motion (0 1), superdiffusive motion (1 2), or no motion at all ( = 0). When few sets of trajectories are available for the ensemble average, time averaged squared displacement (TASD) can be calculated to reduce statistical noise. The TASD of the ith migration path is given by where the overline denotes an average over time t, leaving TASD a function of the time interval, . Squared displacement that is first time averaged and then ensemble averaged, hr2 ti, is hereby referred to as MSD as this is how it is often defined in other studies [14, 17, 26]. Ergodicity, i, is the proportionality factor between time averaged squared displacement of the ith migration path (TASD) and ensemble averaged squared displacement (EASD): A motion is ergodic when the underlying dynamical properties such as cell speed or directional bias do not change over time. Mathematically, this corresponds to = 1 over all time, indicating no difference between the TASD and EASD. This equivalence is useful because TASD measurements can be used to represent EASD measurements when the latter is statistically noisy. In general, time averaging smooths out complexities in the squared displacement caused by variables that change over time (and space), such as instantaneous speed. When such factors change significantly as is often the case with collective and cooperative motion, and motion through confined, spatially heterogeneous topologies (e.g. nonuniform substrates where environmental factors such as adhesiveness vary over time and space), hi 6 1 and the migration paths are said to be nonergodic. Nonergodic motion is mathematically more difficult to characterize and this has implications on directionality time measurements. Characterizing Motion using the Slope of Mean Squared Displacement in log-log Coordinates The slope of EASD plotted in log-log coordinates is an approximate measure of the EASD exponent (Eq 1) and therefore a measure that characterizes trajectory diffusivity and/or directedness. Using Eq 1, and noting that can change with time t, one can define the log-log EASD slope: bt ddat t ln t a. When EASD exponent is constant, = . Otherwise, (t) is an estimator of (t), and therefore an estimator of how diffusive or ballistic the motion is at a particular time, t. In the case where the migration paths are ergodic, EASD and MSD are interchangeable and the log-log MSD slope () can also be calculated to characterize motion as a function of time interval . In this work, the functional form of (t) is mathematically derived and used to determine a sampling interval independent measurement that characterizes directionality. 2D Persistent Biased Random Walk Simulations and Data Fits All simulation data were generated in MATLAB. They were fit using custom MATLAB software. This code is available for download online . Data fits were calculated using the Levenburg-Marquardt least squared fitting algorithm  built into MATLAB. Migration Paths and Centroid Measurement Error Differential interference contrast (DIC) image sequences capturing directionally migrating Polymorphonuclear Human Neutrophils were obtained from Ref.  along with cell centroid positions, ri(tn), where i is the migration path index and tn is the time measured in multiples of the sampling interval, tn = nt (n = 1, 2, . . .). Centroid measurement error was estimated as follows. A cell was chosen at random and manually outlined five times. The five corresponding centroid positions were determined using the regionprops algorithm in MATLAB (the MathWorks; Natick, MA). The centroid measurement error of that cell, m, was calculated as the RMS displacement from the mean centroid position. Deriving the Directionality Time Model When observing an ergodic, directionally biased random walk, there exists a sufficiently large sampling interval such that the motion will appear to be ballistic. Put in terms of log-log MSD slope, () transitions towards 2 as ! 1 for a directionally biased random walk. The idea of using log-log MSD slope to measure the transition time was recently proposed in a preceding article about neutrophil chemotaxis . We suggested an empirical fit function for (t) to quantify this time interval that we called directionality time. Here in this article, we rigorously develop the concept of directionality time from the bottom up by analytically deriving a (t) fit function (without free parameters) and using biased and persistent random walk models to characterize its robustness. Directionality time is defined as the time scale above which motion appears ballistic (directional) and can be loosely interpreted as the time it takes for a random walker to orient towards an external cue. To determine the mathematical meaning and robustness of directionality time, we analytically derive the log-log EASD slope, (t), for three ergodic directionally biased random walk models: 1. Drift Diffusion (DD) 2. 2D Stepping Biased Random Walk (2D-SBRW) 3. 1D Persistent Biased Random Walk in Continuous Time (1D-PBRW) These calculations are shown in detail in the supporting information (Appendix A in S1 File). The high level results are described here. DD (model 1) was a suitable starting point because these processes are readily understood. For DD in d dimensions, with a diffusion constant D, and a drift speed u, the log-log EASD slope is shown in the supporting information (Eq. A2 in S1 File) to be 2t 1 td bDDt 1 ttd where td 2ud2D defines directionality time. Note that (t) begins at (0) = 1 and asymptotes towards 2. This is the signature of a directionally biased random walk. Directionality time is the 3 where the migration transitions from diffusive to directional. As D intime at which b 2 creases and/or u decreases, more observation time is required to determine that the motion is directionally biased. Directional cell migration, though, is not a drift diffusion process. While drift may be an important factor when measuring the directionality of swimming cells, the process of cellular propulsion itself is better described kinematically by a 2D-SBRW (model 2). In a 2D-SBRW process, an object steps from one discrete position, Rn, to the next, Rn+1 (n = 0, 1, 2, . . .), such that the displacements between successive steps are biased towards a particular direction, e^x (Fig 1C). Notationally, all random variables are assigned capital letters. Using Ln and n to denote step lengths and polar angles (orientation with respect to e^x) respectively, the stepwise EASD can be shown to be (see Ref.  for the derivation, and Eq. A5 in S1 File) The important information conveyed by this equation is that the motion is diffusive (hR2ni n) when n is small and directional (hR2ni n2) when n is large. As before, the goal is to derive the directionality time (or number of steps) at which the motion transitions from diffusive to directional. By defining a constant instantaneous speed v, the approximation n hvLti can be used to derive EASD as a function of time t instead of step number n. In the time representation, this is a model of a biased random walk (BRW) instead of an SBRW. Differentiating in log-log coordinates gives the log-log EASD slope (Eq. A8 in S1 File) 2t 1 td bBRWt; td 1 ttd The functional form of EASD slope is no different between models 1 and 2, except that the mathematical constants that constitute directionality time have changed. With Eq 5 holding true for all composite step length and directional bias distributions, this function can be used to measure the directionality time of many types of directionally biased motion. The equivalence between this form of directionality time and that derived for DD (below Eq 4) is shown in Appendix A (Eq. A12 in S1 File). The generalized directionality time given in Eq 6 can be understood by considering the following example. Consider the case where the probability that a cell changes direction is constant from one moment to the next and that orientation angles are chosen from a biased distribution independent of step lengths L. Then directionality time simplifies to (Eq. A13 in S1 File) where c hcosi, and tp represents the reorientation time, the average time of each ballistic step. The term c is tangent-bias correlation (analogous to tangent-tangent correlation). Values of c2 range from 0, corresponding to no orientation bias (PDF rYy 21p, where ), to 1, corresponding to maximal orientation or anti-orientation bias (PDF () = () or ( ), where is the Dirac delta function). Directionality time depends only on the reorientation time and the extent to which the orientation is biased when a reorientation event occurs, increasing with the former and decreasing with the latter. In particular, the term 2 c2c2 ranges from 1 at maximal bias, to 1 at no bias. It may appear odd that td ! tp (the equivalent of an average step duration) when the system is perfectly directional (c2 ! 1). However, this is no more than a subtlety of stepping random walks. There is no change in position defined at t tp because of the way continuous time was substituted in for discrete stepping number (t ntp). Therefore, the minimum time to determine that movement is directionally biased will always be greater than or equal to tp. No information can be gained from a random walker that has not yet taken any steps. In the general case where L is not a Poissonian PDF but stepping time and directional bias are independent of one another, the constant 2 in the numerator of Eq 7 is replaced by hL2i. hLi2 Thus, the directionality time increases with increasing stepping time variance, as one would expect. This is relevant to processes such as Lvy Flights which correspond to a step length distribution with a long tail . Since each step in the SBRW is accompanied by a memoryless reorientation, this model cannot be used to derive a log-log EASD slope equation that accounts for persistence. In order to consider the relation between directionality time and persistence, a continuous time random walk model must be derived, noted as the PBRW (model 3). This model is derived in 1D for simplicity using the biased telegrapher equation (Eq. A14 in S1 File) [24, 30]. To put this model in context, the unbiased telegrapher equation has been used to derive the dynamics and EASD of persistent random walks that describe the kinematics of chemokinesis [16, 31], as well as the motion of grasshoppers and kangaroos . In the 1D-PBRW, an object moves with constant speed v, either left (x direction) or right (+x direction) for some random run time (T(l) or T(r), respectively) before switching directions. Bias is induced by drawing left and right run times from nonequivalent distributions and characterized using tangent-bias correlation c hcosYi hhTTrriihhTTllii (c.f. Eq 7). The log-log EASD slope of the PBRW (Eq. A21 in S1 File) is more complicated than that for the BRW and DD because directionality over short time scales caused by persistence induces a zero-time log-log EASD slope PBRW(0) = 2. As t increases and the orientational correlation of persistent motion is lost, PBRW(t) dips towards 1. Except when c = 0, PBRW(t) ! 2 as t ! 1, as is the signature of directionally biased motion. These PBRW(t) curves are plotted Fig 2A for multiple values of the tangent-bias correlation c2 (solid curves). In this plot, time is in units of l1, which is related to the average run time (persistence time). At sufficiently large time scales (Eq. A22 in S1 File) 2t 1 td bPBRWt tBRW bBRWt; td 1 ttd where tBRW is the convergence time above which the difference between PBRW and BRW is less than 5% (Fig 2B). Therefore, directionality time can be measured by fitting to BRW(t; td) at time scales larger than t = tBRW. The resulting measurement of td from this fit is given by (Eq. A23 in S1 File) As with the BRW, td ! 1 for a random walk that is unbiased. When the bias is sufficient (12 c2 1), the gap between short time scale persistence and long time scale directionality vanishes such that the random walk appears directional at all time scales. By construction, we redefine negative values of directional